Geodesic
Cordial elements and dimensions of affine Deligne–Lusztig varieties
Forum of Mathematics, Pi, Tome 9 (2021)

Voir la notice de l'article provenant de la source Cambridge University Press

The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${\mathbf G}$ depends on two parameters: the $\sigma $-conjugacy class $[b]$ and the element w in the Iwahori–Weyl group $\tilde {W}$ of ${\mathbf G}$. In this paper, for any given $\sigma $-conjugacy class $[b]$, we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde {W}$. 
@article{10_1017_fmp_2021_10,
     author = {Xuhua He},
     title = {Cordial elements and dimensions of affine {Deligne{\textendash}Lusztig} varieties},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {9},
     year = {2021},
     doi = {10.1017/fmp.2021.10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.10/}
}
TY  - JOUR
AU  - Xuhua He
TI  - Cordial elements and dimensions of affine Deligne–Lusztig varieties
JO  - Forum of Mathematics, Pi
PY  - 2021
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.10/
DO  - 10.1017/fmp.2021.10
LA  - en
ID  - 10_1017_fmp_2021_10
ER  - 
%0 Journal Article
%A Xuhua He
%T Cordial elements and dimensions of affine Deligne–Lusztig varieties
%J Forum of Mathematics, Pi
%D 2021
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.10/
%R 10.1017/fmp.2021.10
%G en
%F 10_1017_fmp_2021_10
Xuhua He. Cordial elements and dimensions of affine Deligne–Lusztig varieties. Forum of Mathematics, Pi, Tome 9 (2021). doi: 10.1017/fmp.2021.10

[BS17] Bhatt, B. and Scholze, P., ‘Projectivity of the Witt vector Grassmannian’, Invent. Math. 209 (2017), 329–423.Google Scholar | DOI

[BT72] Bruhat, F. and Tits, J., ‘Groupes rèductifs sur un corps local, I’, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–276.Google Scholar | DOI

[DL76] Deligne, P. and Lusztig, G., ‘Representations of reductive groups over finite fields’, Ann. of Math. (2) 103(1) (1976), 103–161.Google Scholar | DOI

[Ga10] Gashi, Q., ‘On a conjecture of Kottwitz and Rapoport’, Ann. Sci. Éc. Norm. Supér. (4) 43(6) (2010), 1017–1038.Google Scholar | DOI

[GHKR10] Görtz, U., Haines, T., Kottwitz, R. and Reuman, D., ‘Affine Deligne-Lusztig varieties in affine flag varieties’, Compos. Math. 146 (2010), 1339–1382.Google Scholar | DOI

[GH10] Görtz, U. and He, X., ‘Dimension of affine Deligne-Lusztig varieties in affine flag varieties’, Doc. Math. 15 (2010), 1009–1028.Google Scholar

[GHN15] Görtz, U., He, X. and Nie, S., ‘ -alcoves and nonemptiness of affine Deligne-Lusztig varieties’, Ann. Sci. Èc. Norm. Supér. (4) 48 (2015), 647–665.Google Scholar | DOI

[GW10] Görtz, U. and Wedhorn, T., Algebraic Geometry I. Schemes with Examples and Exercises , Advanced Lectures in Mathematics (Vieweg + Teubner, Wiesbaden, Germany, 2010).Google Scholar

[HV11] Hartl, U. and Viehmann, E., ‘The Newton stratification on deformations of local -shtukas’, J. Reine Angew. Math. 656 (2011), 87–129.Google Scholar | DOI

[He09] He, X., ‘A subalgebra of -Hecke algebra’, J. Algebra 322 (2009), 4030–4039.Google Scholar | DOI

[He14] He, X., ‘Geometric and homological properties of affine Deligne-Lusztig varieties’, Ann. of Math. (2) 179 (2014), 367–404.Google Scholar | DOI

[He15] He, X., ‘Hecke algebras and -adic groups’, in Current Developments in Mathematics 2015 (International Press, Somerville, MA, 2016), 73–135.Google Scholar

[HN14] He, X. and Nie, S., ‘Minimal length elements of extended affine Weyl group’, Compos. Math. 150(11) (2014), 1903–1927.Google Scholar | DOI

[HR17] He, X. and Rapoport, M., ‘Stratifications in the reduction of Shimura varieties’, Manuscripta Math. 152 (2017), 317–343.Google Scholar | DOI

[HY12] He, X. and Yang, Z., ‘Elements with finite Coxeter part in an affine Weyl group’, J. Algebra 372 (2012), 204–210.Google Scholar | DOI

[HY21] Yu, Q. and He, X., ‘Dimension formula of the affine Deligne-Lusztig variety ’, Math. Ann. 379 (2021), 1747–1765.Google Scholar

[IM65] Iwahori, N. and Matsumoto, H., ‘On some Bruhat decomposition and the structure of the Hecke rings of -adic Chevalley groups’, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.Google Scholar | DOI

[Ko85] Kottwitz, R., ‘Isocrystals with additional structure’, Compos. Math. 56 (1985), 201–220.Google Scholar

[Ko97] Kottwitz, R., ‘Isocrystals with additional structure. II’, Compos. Math. 109 (1997), 255–339.Google Scholar | DOI

[MST19] Milićević, E., Schwer, P. and Thomas, A., Dimensions of Affine Deligne-Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators, Memoirs of the American Mathematical Society, vol. 261 (2019), no. 1260.Google Scholar | DOI

[MV20] Milićević, E. and Viehmann, E., ‘Generic Newton points and the Newton poset in Iwahori-double cosets’, Forum Math. Sigma 8 (2020), E50.Google Scholar

[Ra05] Rapoport, M., ‘A guide to the reduction modulo of Shimura varieties’, Astérisque 298 (2005), 271–318.Google Scholar

[Vi14] Viehmann, E., ‘Truncations of level 1 of elements in the loop group of a reductive group’, Ann. of Math. (2) 179(3) (2014), 1009–1040.Google Scholar | DOI

[Zu17] Zhu, X., ‘Affine Grassmannians and the geometric Satake in mixed characteristic’, Ann. of Math. (2) 185 (2017), 403–492.Google Scholar | DOI

Cité par Sources :